Optimization-based control design techniques and tools

TL;DR

This paper discusses recent advances in structured output feedback controller synthesis, emphasizing optimization-based methods over traditional algebraic approaches, and highlights software tools for practical implementation.

Contribution

It introduces a new paradigm in structured $H_0$-control synthesis based on optimization techniques, moving away from Riccati equations and matrix inequalities.

Findings

Structured control synthesis is central in modern control engineering.

Non-smooth optimization techniques are essential for computing structured control laws.

Software tools facilitate practical application of these new control design methods.

Abstract

Structured output feedback controller synthesis is an exciting recent concept in modern control design, which bridges between theory and practice in so far as it allows for the first time to apply sophisticated mathematical design paradigms like $H_\infty$- or $H_2$-control within control architectures preferred by practitioners. The new approach to structured $H_\infty$-control, developed by the authors during the past decade, is rooted in a change of paradigm in the synthesis algorithms. Structured design is no longer be based on solving algebraic Riccati equations or matrix inequalities. Instead, optimization-based design techniques are required. In this essay we indicate why structured controller synthesis is central in modern control engineering. We explain why non-smooth optimization techniques are needed to compute structured control laws, and we point to software tools which enable practitioners to use these new tools in high technology applications.

Equations

$$\displaystyle P:\left\{\begin{array}[]{llrlrlr}\dot{x}_{P}&\!\!=\!\!&Ax_{P}&\!\!+\!\!&B_{1}w&\!\!+\!\!&B_{2}u\\ z&\!\!=\!\!&C_{1}x_{P}&\!\!+\!\!&D_{11}w&\!\!+\!\!&D_{12}u\\ y&\!\!=\!\!&C_{2}x_{P}&\!\!+\!\!&D_{21}w&\!\!+\!\!&D_{22}u\end{array}\right.$$

$$\displaystyle K:\left\{\begin{array}[]{clrlr}\dot{x}_{K}&\!\!\!=\!\!&A_{K}x_{K}&\!\!+\!\!&B_{K}y\\ u&\!\!\!=\!\!&C_{K}x_{K}&\!\!+\!\!&D_{K}y\end{array}\right.$$

$$A_K=A_K(\mathbf{x}),B_K=B_K(\mathbf{x}),C_K=C_K(\mathbf{x}),D_K=D_K(\mathbf{x}),$$

$$\displaystyle\begin{array}[]{ll}\mbox{minimize}&\|T_{wz}(P,K(\mathbf{x}))\|\\ \mbox{subject to}&\mbox{$K(\mathbf{x})$ closed-loop stabilizing}\\ &\mbox{$K(\mathbf{x})$ \mbox{structured}, \,$\mathbf{x}\in\mathbb{R}^{n}$ }\end{array}$$

$$\displaystyle\!\!\!\!\!\!\begin{array}[]{c}\left[\begin{array}[]{cc|c}0&0&1\\ 0&-1/\tau&-k_{D}/\tau\\ \hline\cr k_{I}&1/\tau&k_{P}+k_{D}/\tau\end{array}\right],\left[\begin{array}[]{c|c}A-B_{2}K_{c}-K_{f}C_{2}&K_{f}\\ \hline\cr-K_{c}&0\end{array}\right],\\ \left[\begin{array}[]{c|c}A_{K}&B_{K}\\ \hline\cr C_{K}&D_{K}\end{array}\right],\\ \left[\begin{array}[]{c|c}{\rm diag}(A_{K_{1}},\ldots,A_{K_{q}})&{\rm diag}(B_{K_{1}},\ldots,B_{K_{q}})\\ \hline\cr{\rm diag}(C_{K_{1}},\ldots,C_{K_{q}})&{\rm diag}(D_{K_{1}},\ldots,D_{K_{q}})\\ \end{array}\right].\end{array}$$

$$\displaystyle P:=\left[\begin{array}[]{c|ccc}A&0&0&B\\ \hline\cr C&0&0&D\\ 0&I&0&0\\ -C&0&I&-D\end{array}\right],$$

$$\displaystyle K(\mathbf{x},s):=\left[\begin{array}[]{cc}K_{1}(a,s)&0\\ 0&K_{2}(s,k_{P},k_{I})\\ \end{array}\right]\,,$$

$$\displaystyle\begin{array}[]{ll}\mbox{minimize}&f(\mathbf{x})=\max\limits_{k\in{\rm SOFT},i\in I_{k}}\|T_{w_{i}z_{i}}^{(k)}(K(\mathbf{x}))\|\\ \mbox{subject to}&g(\mathbf{x})=\max\limits_{k\in{\rm HARD},j\in J_{k}}\|T_{w_{j}z_{j}}^{(k)}(K(\mathbf{x}))\|\leq 1\\ &K(\mathbf{x})\mbox{ structured and stabilizing}\\ &\mathbf{x}\in\mathbb{R}^{n}\end{array}$$

$$\displaystyle\begin{array}[]{ll}\mbox{minimize}&f(\mathbf{x})\\ \mbox{subject to}&g(\mathbf{x})\leq 0\\ &\mathbf{x}\in\mathbb{R}^{n}\end{array}$$

$$x_{tt}(\xi ,t)$$

$$x_{\xi }(0,t)$$

$$x_{\xi }(1,t)$$

$$y_1(t)=x(0,t),y_2(t)=x(1,t),y_3(t)=x_t(1,t).$$

$$G(\xi ,s)=\frac{x(\xi ,s)}{u(s)}=\frac{1}{s}\cdot \frac{(1-q)e^{s\xi }+(1+q)e^{-s\xi }}{(1-q)e^s-(1+q)e^{-s}},$$

$$\hat{G}(s)=\left[\begin{array}{c}\frac{1}{s(1-q)}\text{vspace}.2cm\\ \frac{1+Q}{2s}\text{vspace}.2cm\\ \frac{1}{2}\end{array}\right]+\left[\begin{array}{c}-\frac{1-e^{-s}}{s(1-q)}\text{vspace}.2cm\\ -\frac{Q(1-e^{-2s})}{2s}\\ \text{vspace}.2cm\frac{Q}{2}{e}^{-2s}\end{array}\right]=:\tilde{G}(s)+\Phi (s).$$

$$\tilde{K}(q,\mathbf{x})=\tilde{K}(q_0)+(q-q_0){\tilde{K}}_1(\mathbf{x})+(q-q_0{)}^2{\tilde{K}}_2(\mathbf{x}),$$

$$\tilde{K}(q,{\mathbf{x}}^{\ast })=\tilde{K}(q_0)+(q-q_0){\tilde{K}}_1({\mathbf{x}}^{\ast })+(q-q_0{)}^2{\tilde{K}}_2({\mathbf{x}}^{\ast }),$$

$$K^{(2)}(q)=K_0+\mathtt{feedback}(\tilde{K}(q,{\mathbf{x}}^{\ast }),-\Phi (q)).$$

Full text

Keywords and Phrases Controller tuning, $H_{\infty}$ synthesis, multi-objective design, nonsmooth optimization, structured controllers, robust control

Pierre Apkarian is with ONERA, 2 Av. Ed. Belin, 31055, Toulouse, France Pierre.Apkarian@onera.fr

Dominikus Noll is with Université Paul Sabatier, Institut de Mathématiques, 118, route de Narbonne, 31062 Toulouse, France dominikus.noll@math.univ-toulouse.fr

Introduction

In the modern high technology field control engineers usually face a large variety of concurring design specifications such as noise or gain attenuation in prescribed frequency bands, damping, decoupling, constraints on settling- or rise-time, and much else. In addition, as plant models are generally only approximations of the true system dynamics, control laws have to be robust with respect to uncertainty in physical parameters or with regard to un-modeled high frequency phenomena. Not surprisingly, such a plethora of constraints presents a major challenge for controller tuning, due not only to the ever growing number of such constraints, but also because of their very different provenience.

The steady increase in plant complexity is exacerbated by the quest that regulators should be as simple as possible, easy to understand and to tune by practitioners, convenient to hardware implement, and generally available at low cost. These practical constraints highlight the limited use of Riccati- or LMI-based controllers, and they are the driving force for the implementation of structured control architectures. On the other hand this means that hand-tuning methods have to be replaced by rigorous algorithmic optimization tools.

1. Structured Controllers

Before addressing specific optimization techniques, we recall some basic terminology for control design problems with structured controllers. The plant model $P$ is described as

[TABLE]

where $A$, $B_{1}$, … are real matrices of appropriate dimensions, $x_{P}\in\mathbb{R}^{n_{P}}$ is the state, $u\in\mathbb{R}^{n_{u}}$ the control, $y\in\mathbb{R}^{n_{y}}$ the measured output, $w\in\mathbb{R}^{n_{w}}$ the exogenous input, and $z\in\mathbb{R}^{n_{z}}$ the regulated output. Similarly, the sought output feedback controller $K$ is described as

[TABLE]

with $x_{K}\in\mathbb{R}^{n_{K}}$, and is called structured if the (real) matrices $A_{K},B_{K},C_{K},D_{K}$ depend smoothly on a design parameter $\mathbf{x}\in\mathbb{R}^{n}$, referred to as the vector of tunable parameters. Formally, we have differentiable mappings

[TABLE]

and we abbreviate these by the notation $K(\mathbf{x})$ for short to emphasize that the controller is structured with $\mathbf{x}$ as tunable elements. A structured controller synthesis problem is then an optimization problem of the form

[TABLE]

where $T_{wz}(P,K)=\mathcal{F}_{\ell}(P,K)$ is the lower feedback connection of (4) with (7) as in Fig. 1 (left), also called the Linear Fractional Transformation [1]. The norm $\|\cdot\|$ stands for the $H_{\infty}$-norm, the $H_{2}$-norm, or any other system norm, while the optimization variable $\mathbf{x}\in\mathbb{R}^{n}$ regroups the tunable parameters in the design.

Standard examples of structured controllers $K(\mathbf{x})$ include realizable PIDs, observer-based, reduced-order, or decentralized controllers, which in state-space are expressed as:

[TABLE]

In the case of a PID the tunable parameters are $\mathbf{x}=(\tau,k_{P},k_{I},k_{D})$, for observer-based controllers $\mathbf{x}$ regroups the estimator and state-feedback gains $(K_{f},K_{c})$, for reduced order controllers $n_{K}<n_{P}$ the tunable parameters $\mathbf{x}$ are the $n_{K}^{2}+n_{K}n_{y}+n_{K}n_{u}+n_{y}n_{u}$ unknown entries in $(A_{K},B_{K},C_{K},D_{K})$, and in the decentralized form $\mathbf{x}$ regroups the unknown entries in $A_{K1},\dots,D_{Kq}$. In contrast, full-order controllers have the maximum number $N=n_{P}^{2}+n_{P}n_{y}+n_{P}n_{u}+n_{y}n_{u}$ of degrees of freedom and are referred to as unstructured or as black-box controllers.

More sophisticated controller structures $K(\mathbf{x})$ arise form architectures like for instance a 2-DOF control arrangement with feedback block $K_{2}$ and a set-point filter $K_{1}$ as in Fig. 1 (right). Suppose $K_{1}$ is the $1$st-order filter $K_{1}(s)=a/(s+a)$ and $K_{2}$ the PI feedback $K_{2}(s)=k_{P}+k_{I}/s$. Then the transfer $T_{ry}$ from $r$ to $y$ can be represented as the feedback connection of $P$ and $K(\mathbf{x},s)$ with

[TABLE]

and gathering tunable elements in $\mathbf{x}=(a,k_{P},k_{I})$.

In much the same way arbitrary multi-loop interconnections of fixed-model elements with tunable controller blocks $K_{i}(\mathbf{x})$ can be re-arranged as in Fig. 2, so that $K(\mathbf{x})$ captures all tunable blocks in a decentralized structure general enough to cover most engineering applications.

The structure concept is equally useful to deal with the second central challenge in control design: system uncertainty. The latter may be handled with $\mu$-synthesis techniques [2] if a parametric uncertain model is available. A less ambitious but often more practical alternative consists in optimizing the structured controller $K(\mathbf{x})$ against a finite set of plants $P^{(1)},\dots,P^{(M)}$ representing model variations due to uncertainty, aging, sensor and actuator breakdown, un-modeled dynamics, in tandem with the robustness and performance specifications. This is again formally covered by Fig. 2 and leads to a multi-objective constrained optimization problem of the form

[TABLE]

where $T_{w_{i}z_{i}}^{(k)}$ denotes the $i$th closed-loop robustness or performance channel $w_{i}\to z_{i}$ for the $k$-th plant model $P^{(k)}$. SOFT and HARD denote index sets taken over a finite set of specifications, say in $\left\{1,\ldots,M\right\}$. The rationale of (22) is to minimize the worst-case cost of the soft constraints $\|T_{w_{i}z_{i}}^{(k)}\|$, $k\in\,$SOFT, while enforcing the hard constraints $\|T_{w_{j}z_{j}}^{(k)}\|\leq 1$, $k\in\,$HARD, which prevail over soft ones and are mandatory. In addition to local optimization (22), the problem can undergo a global optimization step in order to prove global stability and performance of the design, see [3, 4, 5].

2. Optimization Techniques Over the Years

During the late 1990s the necessity to develop design techniques for structured regulators $K(\mathbf{x})$ was recognized [6], and the limitations of synthesis methods based on algebraic Riccati equations or linear matrix inequalities (LMIs) became evident, as these techniques cannot provide structured controllers needed in practice. The lack of appropriate synthesis techniques for structured $K(\mathbf{x})$ led to the unfortunate situation, where sophisticated approaches like the $H_{\infty}$ paradigm developed by academia since the 1980s could not be brought to work for the design of those controller structures $K(\mathbf{x})$ preferred by practitioners. Design engineers had to continue to rely on heuristic and ad-hoc tuning techniques, with only limited scope and reliability. As an example: post-processing to reduce a black-box controller to a practical size is prone to failure. It may at best be considered a fill-in for a rigorous design method which directly computes a reduced-order controller. Similarly, hand-tuning of the parameters $\mathbf{x}$ remains a puzzling task because of the loop interactions, and fails as soon as complexity increases.

In the late 1990s and early 2000s, a change of methods was observed. Structured $H_{2}$- and $H_{\infty}$-synthesis problems (11) were addressed by bilinear matrix inequality (BMI) optimization, which used local optimization techniques based on the augmented Lagrangian method [6, 7, 8, 9], sequential semidefinite programming methods [10, 11], and non-smooth methods for BMIs [12, 13]. However, these techniques were based on the bounded real lemma or similar matrix inequalities, and were therefore of limited success due to the presence of Lyapunov variables, i.e. matrix-valued unknowns, whose dimension grows quadratically in $n_{P}+n_{K}$ and represents the bottleneck of that approach.

The epoch-making change occurs with the introduction of non-smooth optimization techniques [14, 15, 16, 17] to programs (11) and (22). Today non-smooth methods have superseded matrix inequality-based techniques and may be considered the state-of-art as far as realistic applications are concerned. The transition took almost a decade.

Alternative control-related local optimization techniques and heuristics include the gradient sampling technique of [18], and other derivative-free optimization as in [19, 20], particle swarm optimization, see [21] and references therein, and also evolutionary computation techniques [22]. All these classes do not exploit derivative information and rely on function evaluations only. They are therefore applicable to a broad variety of problems including those where function values arise from complex numerical simulations. The combinatorial nature of these techniques, however, limits their use to small problems with a few tens of variable. More significantly, these methods often lack a solid convergence theory. In contrast, as we have demonstrated over recent years, [15, 23, 24, 25] specialized non-smooth techniques are highly efficient in practice, are based on a sophisticated convergence theory, capable of solving medium size problems in a matter of seconds, and are still operational for large size problems with several hundreds of states.

3. Non-smooth optimization techniques

The benefit of the non-smooth casts (11) and (22) lies in the possibility to avoid searching for Lyapunov variables, a major advantage as their number $(n_{P}+n_{K})^{2}/2$ usually largely dominates $n$, the number of true decision parameters $\mathbf{x}$. Lyapunov variables do still occur implicitly in the function evaluation procedures, but this has no harmful effect for systems up to several hundred states. In abstract terms, a non-smooth optimization program has the form

[TABLE]

where $f,g:\mathbb{R}^{n}\to\mathbb{R}$ are locally Lipschitz functions and are easily identified from the cast in (22).

In the realm of convex optimization, non-smooth programs are conveniently addressed by so-called bundle methods, introduced in the late 1970s by Lemaréchal [26]. Bundle methods are used to solve difficult problems in integer programming or in stochastic optimization via Lagrangian relaxation. Extensions of the bundling technique to non-convex problems like (11) or (22) were first developed in [15, 16, 17, 27, 12], and in more abstract form, in [23]. Recently, we also extended bundle techniques to the trust-region framework [24, 25, 28], which leads to the first extension of the classical trust-region method to non-differential optimization supported by a valid convergence theory.

Fig. 3 shows a schematic view of a non-convex bundle method consisting of a descent-step generating inner loop (yellow block) comparable to a line search in smooth optimization, embedded into the outer loop (blue box), where serious iterates are processed, stopping criteria are applied, and the acceptance rules of traditional trust region techniques is assured. At the core of the interaction between inner and outer loop is the management of the proximity control parameter $\tau$, which governs the stepsize $\|\mathbf{x}-\mathbf{y}^{k}\|$ between trial steps $\mathbf{y}^{k}$ at the current serious iterate $\mathbf{x}$. Similar to the management of a trust region radius or of the stepsize in a linesearch, proximity control allows to force shorter trial steps if agreement of the local model with the true objective function is poor, and allows larger steps if agreement is satisfactory.

Oracle-based bundle methods traditionally assure global convergence in the sense of subsequences under the sole hypothesis that for every trial point $\mathbf{x}$ the function value $f(\mathbf{x})$ and one Clarke subgradient $\phi\in\partial f(\mathbf{x})$ are provided. In automatic control applications it is as a rule possible to provide more specific information, which may be exploited to speed up convergence [15].

Computing function value and gradients of the $H_{2}$-norm $f(\mathbf{x})=\|T_{wz}\left(P,K(\mathbf{x})\right)\|_{2}$ requires essentially the solution of two Lyapunov equations of size $n_{P}+n_{K}$, see [29]. For the $H_{\infty}$-norm, $f(\mathbf{x})=\|T_{wz}\left(P,K(\mathbf{x})\right)\|_{\infty}$, function evaluation is based on the Hamiltonian algorithm of [30, 31]. The Hamiltonian matrix is of size $n_{P}+n_{K}$, so that function evaluations may be costly for very large plant state dimension ($n_{P}>500$), even though the number of outer loop iterations of the bundle algorithm is not affected by a large $n_{P}$ and generally relates to $n$, the dimension of $\mathbf{x}$. The additional cost for subgradient computation for large $n_{P}$ is relatively cheap as it relies on linear algebra [15]. Function and subgradient evaluations for $H_{\infty}$ and $H_{2}$ norms are typically obtained in $\mathcal{O}\left((n_{P}+n_{K})^{3}\right)$ flops.

4. Computational Tools

Our non-smooth optimization methods became available to the engineering community since 2010 via the MATLAB Robust Control Toolbox [32, 33]. Routines hinfstruct, looptune  and systune  are versatile enough to define and combine tunable blocks $K_{i}(\mathbf{x})$, to build and aggregate multiple models and design requirements on $T_{wz}^{(k)}$ of different nature, and to provide suitable validation tools. Their implementation was carried out in cooperation with P. Gahinet (MathWorks). These routines further exploit the structure of problem (22) to enhance efficiency, see [16] and [15].

It should be mentioned that design problems with multiple hard constraints are inherently complex and generally NP-hard, so that exhaustive methods fail even for small to medium size problems. The principled decision made in [15], and reflected in the MATLAB tools, is to rely on local optimization techniques instead. This leads to weaker convergence certificates, but has the advantage to work successfully in practice. In the same vein, in (22) it is preferable to rely on a mixture of soft and hard requirements, for instance, by the use of exact penalty functions [14]. Key features implemented in the mentioned MATLAB routines are discussed in [34, 33, 16].

5. Applications

Design of a feedback regulator is an interactive process, in which tools like systune, looptune   or hinfstruct   support the designer in various ways. In this section we illustrate their enormous potential by showing that even infinite-dimensional systems may be successfully addressed by these tools. For a plethora of design examples for real-rational systems including parametric and complex dynamic uncertainty we refer to [3, 24, 4, 28, 25]. For recent applications of our tools in real-world applications see also [35], where it is in particular explained how hinfstruct helped in 2014 to save the Rosetta mission. Another important application of hinfstruct is the design of the atmospheric flight pilot for the ARIANE VI launcher by the Ariane Group [36].

5.1. Illustrative example

We discuss boundary control of a wave equation with anti-stable damping,

[TABLE]

where notations $x_{t}$, $x_{\xi}$ are partial derivatives of $x$ with respect to time and space, respectively. In (5.1), $x(\cdot,t),x_{t}(\cdot,t)$ is the state, the control applied at the boundary $\xi=1$ is $u(t)$, and we assume that the measured outputs are

[TABLE]

The system has been discussed previously in [37, 38, 39] and has been proposed for the control of slip-strick vibrations in drilling devices [40]. Here measurements $y_{1},y_{2}$ correspond to the angular positions of the drill string at the top and bottom level, $y_{3}$ measures angular speed at the top level, while control corresponds to a reference velocity at the top. The friction characteristics at the bottom level are characterized by the parameter $q$, and the control objective is to maintain a constant angular velocity at the bottom.

Similar models have been used to control pressure fields in duct combustion dynamics, see [41]. The challenge in (5.1), (28) is to design implementable controllers despite the use of an infinite-dimensional system model.

The transfer function of (5.1) is obtained from:

[TABLE]

which in view of (28) leads to $G(s)=[G_{1};G_{2};G_{3}]=[G(0,s);G(1,s);sG(1,s)]$.

Putting $G$ in feedback with the controller $K_{0}=[0\;0\;1]$ leads to $\widehat{G}=G/(1+G_{3})$, where

[TABLE]

Here $\widetilde{G}$ is real-rational and unstable, while $\Phi$ is stable but infinite dimensional. Now we use the fact that stability of the closed loop $(\widetilde{G}+\Phi,K)$ is equivalent to stability of the loop $(\widetilde{G},{\tt feedback}(K,\Phi))$ upon defining ${\tt feedback}(M,N):=M(I+NM)^{-1}\,.$ The loop transformation is explained in Fig. 4, see also [42].

Using (29) we construct a finite-dimensional structured controller $\widetilde{K}=\widetilde{K}(\mathbf{x})$ which stabilizes $\widetilde{G}$. The controller $K$ stabilizing $\widehat{G}$ in (29) is then recovered from $\widetilde{K}$ through the equation $\widetilde{K}={\tt feedback}(K,\Phi)$, which when inverted gives $K={\tt feedback}(\widetilde{K},-\Phi)$. The overall controller for (5.1) is $K^{*}=K_{0}+K$, and since along with $K$ only delays appear in $\Phi$, the controller $K^{*}$ is implementable.

Construction of $\widetilde{K}$ uses systune  with pole placement via TuningGoal.Poles, imposing that closed-loop poles have a minimum decay of 0.9, minimum damping of 0.9, and a maximum frequency of 4.0. The controller structure is chosen as static, so that $\mathbf{x}\in\mathbb{R}^{3}$. A simulation with $K^{*}$ is shown in Fig. 5 (bottom) and some acceleration over the backstepping controller from [39] (top) is observed.

5.2. Gain-scheduling control

Our last study is when the parameter $q\geq 0$ is uncertain or allowed to vary in time with sufficiently slow variations as in [43]. We assume that a nominal $q_{0}>0$ and an uncertain interval $[\underline{q},\overline{q}]$ with $q_{0}\in(\underline{q},\overline{q})$ and $1\not\in[\underline{q},\overline{q}]$ are given.

The following scheduling scenarios, all leading to implementable controllers, are possible. (a) Computing a nominal controller $\widetilde{K}$ at $q_{0}$ as before, and scheduling through $\Phi(q)$, which depends explicitly on $q$, so that $K^{(1)}(q)=K_{0}+{\tt feedback}(\widetilde{K},-\Phi(q))$. (b) Computing $\widetilde{K}(q)$ which depends on $q$, and using $K^{(2)}(q)=K_{0}+{\tt feedback}(\widetilde{K}(q),-\Phi(q))$.

While (a) uses (11) based on [15, 17] and available in systune, we show that one can also apply (11) to case (b). We use Fig. 4 to work in the finite-dimensional system $(\widetilde{G}(q),\widetilde{K}(q))$, where plant and controller now depend on $q$, which is a parameter-varying design.

For that we have to decide on a parametric form of the controller $\widetilde{K}(q)$, which we choose as

[TABLE]

and where we adopted the simple static form $\widetilde{K}_{1}(\mathbf{x})=[\mathbf{x}_{1}\;\mathbf{x}_{2}\;\mathbf{x}_{3}],\widetilde{K}_{2}=[\mathbf{x}_{4}\;\mathbf{x}_{5}\;\mathbf{x}_{6}]$, featuring a total of 6 tunable parameters. The nominal $\widetilde{K}(q_{0})$ is obtained via (11) as above. For $q_{0}=3$ this leads to $\widetilde{K}(q_{0})=[-1.049\;-1.049\;-0.05402]$, computed via systune.

With the parametric form $\widetilde{K}(q,\mathbf{x})$ fixed, we now use again the feedback system $(\widetilde{G}(q),\widetilde{K}(q))$ in Fig. 4 and design a parametric robust controller using the method of [4], which is included in the systune  package and used by default if an uncertain closed-loop is entered. The tuning goals are chosen as constraints on closed-loop poles including minimum decay of 0.7, minimum damping of 0.9, with maximum frequency $2$. The controller obtained is (with $q_{0}=3$)

[TABLE]

with $\widetilde{K}_{1}=[-0.1102,-0.1102,-0.1053]$, $\widetilde{K}_{2}=[0.03901,0.03901,0.02855]$, and we retrieve the final parameter varying controller for $G(q)$ as

[TABLE]

Nominal and scheduled controllers are compared in simulation in Figs. 6, 7, and 8, which indicate that $K^{(2)}(q)$ achieves the best performance for frozen-in-time values $q\in[2,4]$. All controllers are easily implementable, since only real-rational elements in combination with delays are used.

The non-smooth program (26) was solved with systune  in $30$s CPU on a Mac OS X with $2.66$ GHz Intel Core i7 and $8$ GB RAM. The reader is referred to the MATLAB Control Toolbox 2018b and higher versions for additional examples. More details on this study can be found in [44].